If $G$ is a connected graph then its center is the vertex $v$ such that the maximum of distances from $v$ to the other vertices of $G$ is minimal possible. Prove that a tree has either one center or two adjacent centers. Give an example of a tree of each type with $7$ vertices.

2 Answers
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The chain graph shown above has only one centre, $d$. The largest distance from $d$ to any other vertex is $3$, from $d$ to $a$ and from $d$ to $g$. The largest distance from $c$ to any other vertex is $4$, to vertex $g$, which is greater than $3$, so $c$ is not a centre. Similarly, the greatest distance from $e$ to any other vertex is $4$, to $a$, and you can check that the other vertices are even worse.

a
\
\
b---c---d---e---f
/
/
g

The tree above has two centres; can you find them?

HINT for the proof: Let $T$ be a tree. If $u$ and $v$ are any vertices of $T$, there is a unique path from $u$ to $v$ in $T$. (You’ve probably proved this already; if not, you’ll want to do so now.) Call the length of this path the distance between $u$ and $v$ in $T$. Among all pairs of vertices of $T$ pick two, $u$ and $v$, with the largest possible distance between them. If that distance is even, there’s a vertex smack in the middle of that path; prove that it’s the unique centre of $T$. If the distance between $u$ and $v$ is odd, the path looks, for example, like this:

a---b---c---d---e---f

Now there is no vertex right at the centre of the path, but there are two, $c$ and $d$, that are closest to the centre; prove that those two vertices are the centres of $T$.

@Jane: The maximum distance from $b$ to another vertex is $4$, to $f$; the maximum distance from $d$ to another vertex, however, is only $3$, to $a$ or $g$. Thus, only one of them can possibly be a centre; which one? And when you’ve answered that, what vertex is the other centre?
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Brian M. ScottFeb 14 '13 at 6:15

Ok got it thanks for clearing this up. The two centers are c and d
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Jane KeFeb 14 '13 at 6:20